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Triangle read by rows: T(n,k) = number of labeled bicolored nonseparable graphs with k points in one color class and n-k points in the other class. The classes are interchangeable if k = n-k. Here n >= 2, k=1..n-1.
3

%I #9 Jan 04 2021 18:21:37

%S 1,0,0,0,3,0,0,10,10,0,0,15,340,15,0,0,21,6965,6965,21,0,0,28,51296,

%T 246295,51296,28,0,0,36,326676,14750946,14750946,326676,36,0,0,45,

%U 1917840,322476210,796058676,322476210,1917840,45,0,0,55,10683255

%N Triangle read by rows: T(n,k) = number of labeled bicolored nonseparable graphs with k points in one color class and n-k points in the other class. The classes are interchangeable if k = n-k. Here n >= 2, k=1..n-1.

%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

%H Andrew Howroyd, <a href="/A123474/b123474.txt">Table of n, a(n) for n = 2..1276</a> (first 50 rows; first 24 rows from R. W. Robinson)

%H F. Harary and R. W. Robinson, <a href="http://dx.doi.org/10.4153/CJM-1979-007-3">Labeled bipartite blocks</a>, Canad. J. Math., 31 (1979), 60-68.

%F From _Andrew Howroyd_, Jan 03 2021: (Start)

%F T(n,k) = f(n-2*k) * binomial(n,k) * A123301(n, k) where f(0) = 1/2 and 1 otherwise.

%F A004100(n) = Sum_{k=0..floor(n/2)} T(n,k). (End)

%e Triangle begins:

%e 1;

%e 0, 0;

%e 0, 3, 0;

%e 0, 10, 10, 0;

%e 0, 15, 340, 15, 0;

%e 0, 21, 6965, 6965, 21, 0;

%e 0, 28, 51296, 246295, 51296, 28, 0;

%e ...

%e Formatted as an array:

%e ==========================================================

%e m/n | 1 2 3 4 5 6

%e ----+-----------------------------------------------------

%e 1 | 1 0 0 0 0 0 ...

%e 2 | 0 3 10 15 21 28 ...

%e 3 | 0 10 340 6965 51296 326676 ...

%e 4 | 0 15 6965 246295 14750946 322476210 ...

%e 5 | 0 21 51296 14750946 796058676 105725374062 ...

%e 6 | 0 28 326676 322476210 105725374062 9736032295374 ...

%e ...

%Y Central coefficients are A005335.

%Y Cf. A004100, A123301, A262307.

%K nonn,tabl

%O 2,5

%A _N. J. A. Sloane_, Nov 12 2006